Regularity Criterion for the 3D Navier–Stokes Equations in the Boardline Case.

By developing the local energy bound with B ˙ ∞ , ∞ - 1 and fully using the property of B ˙ ∞ , c (N) - 1 and B ˙ p , ∞ - 1 + 3 / p ∩ B ˙ ∞ , q - 1 (p , q < ∞) , we provide a blow-up criterion to the 3D Navier–Stokes equations in the critical Besov space with - 1 negative regularity. More precisely, if u 3 ∈ L ∞ (0 , T ; B ˙ p , ∞ - 1 + 3 / p ∩ B ˙ ∞ , q - 1) and u h ∈ L ∞ (0 , T ; B ˙ ∞ , ∞ - 1) with u h (T) ∈ B ˙ ∞ , c (N) - 1 , the smooth solution u = (u h , u 3) can be extended beyond T, which improves the result of Wang-Zhang (Sci China Math 60: 637–650, 2017) and Li-Zhou (Dyn PDE 18: 71–80, 2021). [ABSTRACT FROM AUTHOR]